# Second noble kipiscoidal icositetrahedron

Second noble kipiscoidal icositetrahedron
Rank3
TypeNoble
Elements
Faces24 irregular pentagons
Edges12+12+12+12+12
Vertices24
Vertex figureIrregular pentagon
Related polytopes
ArmyNonuniform snub cube
DualSecond noble kisombreroidal icositetrahedron
Convex coreNon-Catalan pentagonal icositetrahedron
Abstract & topological properties
Flag count240
Euler characteristic–12
OrientableNo
Genus14
Properties
SymmetryB3+, order 24
Flag orbits10
ConvexNo
NatureTame

The second noble kipiscoidal icositetrahedron is a noble polyhedron. Its 24 congruent faces are irregular pentagons meeting at congruent order-5 vertices. It is a faceting of a non-uniform snub cubic hull.

The ratio between the longest and shortest edges is 1:2.68320.

## Vertex coordinates

This polyhedron has coordinates given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes, of

• (1, a, b),

where

• ${\displaystyle a={\sqrt[{3}]{1+{\sqrt {\frac {19}{27}}}}}+{\sqrt[{3}]{1-{\sqrt {\frac {19}{27}}}}}+1\approx 2.76929}$

is the real root of ${\displaystyle x^{3}-3x^{2}+x-1}$, and

• ${\displaystyle b={\sqrt[{3}]{{\frac {46}{27}}+{\sqrt {\frac {76}{27}}}}}+{\sqrt[{3}]{{\frac {46}{27}}-{\sqrt {\frac {76}{27}}}}}+{\frac {1}{3}}\approx 2.13039}$

is the real root of ${\displaystyle x^{3}-x^{2}-x-3}$.

These are the same coordinates as the noble kipentagrammic icositetrahedron.